Continuum Mechanics & Field Models

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Abstractions about named mathematical models and formulas drawn from continuum mechanics, fluid dynamics, and geometry, including elasticity and rheology laws (Glen–Nye, Gent, Yeoh), fluid phenomena (Rankine vortex, Moffatt eddies, hydrostatic equilibrium), and coordinate or surface constructions.

42 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Airy function — In mathematics, the Airy function (or Airy function of the first kind) \mathbf{Ai(\boldsymbol x)} is a special function named after the British astronomer George Biddell Airy.
• Central potential — Central forces that are conservative can always be expressed as the negative gradient of a potential energy.
• Channel surface — In geometry and topology, a channel surface or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its directrix.
• Coons patch — In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains into elements.
• Differential Inclusion — A continuous-time evolution law that permits a state-dependent set of instantaneous velocities rather than prescribing just one.
• Earnshaw paradox — In fluid dynamics, the Earnshaw paradox is a physical paradox related to considering sound waves in an ideal inviscid fluid.
• Engquist–Majda absorbing boundary condition — In numerical methods for partial differential equations, Engquist–Majda absorbing boundary conditions or Lindman–Engquist–Majda absorbing boundary conditions are a hierarchy of absorbing boundary conditions for the numerical solution of wave equations.
• Finite strain theory — In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory.
• Gardner's relation — Gardner, is an empirically derived equation that relates seismic P-wave velocity to the bulk density of the lithology in which the wave travels.
• Gent hyperelastic model — The Gent hyperelastic material model is a phenomenological model of rubber elasticity that is based on the concept of limiting chain extensibility.
• Glen–Nye flow law — In theoretical glaciology and continuum mechanics, the Glen–Nye flow law, also referred to as Glen's flow law, is an empirically derived constitutive relation widely used as a model for the rheology of glacial ice.
• Homoclinic bifurcation — A homoclinic bifurcation is a global change in a dynamical system caused when a periodic orbit or invariant manifold forms or loses a trajectory connecting a saddle point to itself.
• Hybrid difference scheme — The hybrid difference scheme is a method used in the numerical solution for convection–diffusion problems.
• Hydrostatic equilibrium — In fluid mechanics, hydrostatic equilibrium, also called hydrostatic balance and hydrostasy, is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force.
• Inertial manifold — In mathematics, inertial manifolds are concerned with the long term behavior of the solutions of dissipative dynamical systems.
• Lagrangian Ocean Analysis — Lagrangian ocean analysis is a way of analysing ocean dynamics by computing the trajectories of virtual fluid particles, following the Lagrangian perspective of fluid flow, from a specified velocity field.
• Laplace expansion (potential) — In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance ( 1 / r ), such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials.
• Linear elasticity — Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions.
• Maslov index — Maslov index is an integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis.
• Modal Analysis — Modal analysis is the study of the dynamic properties of systems in the frequency domain.
• Moffatt eddies — Moffatt eddies are sequences of eddies that develop in corners bounded by plane walls (or sometimes between a wall and a free surface) due to an arbitrary disturbance acting at asymptotically large distances from the corner.
• Momentum — Capture a system's quantity of motion as the conserved additive vector p = mv, so any interaction — however messy inside — collapses to an algebraic balance where the total before equals the total after up to external impulse.
• Monopole moduli space — In mathematics, the monopole moduli space is a space parametrizing monopoles (solutions of the Bogomolny equations). studied the moduli space for 2 monopoles in detail and used it to describe the scattering of monopoles.
• Napkin ring problem — In geometry, the napkin-ring problem involves finding the volume of what remains after a circular hole is drilled through a sphere.
• Near-field (mathematics) — In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.
• Nome (mathematics) — In mathematics, specifically the theory of elliptic functions, the nome is a special function that belongs to the non-elementary functions.
• Oblate Spheroidal Coordinates — Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci.
• Partial differential algebraic equation — In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.
• Poisson geometry — In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure.
• Prolate Spheroidal Coordinates — Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located.
• Quadrature domains — It is known that quadrature domains exist for all values of k.
• Rankine vortex — The Rankine vortex is a simple mathematical model of a vortex in a viscous fluid.
• Right-Hand Rule — In mathematics and physics, the right-hand rule is a convention and a mnemonic utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic field.
• Shields formula — The Shields formula is a formula for the stability calculation of granular material (sand, gravel) in running water.
• Solid of revolution — In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary).
• Stokes's law — {\vec F}_{\rm d} is the frictional force – known as Stokes's drag – acting on the interface between the fluid and the particle (newtons, kg m s −2 ).
• Surface-area-to-volume ratio — The surface-area-to-volume ratio or surface-to-volume ratio (denoted as SA:V, SA/V, or sa/vol) is the ratio between surface area and volume of an object or collection of objects.
• Time-translation symmetry — Time-translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval.
• Torricelli's Equation — In physics, Torricelli's equation, or Torricelli's formula, is an equation created by Evangelista Torricelli to find the final velocity of a moving object with constant acceleration along an axis (for example, the x axis) without having a known time interval.
• Transformation of the electromagnetic field — In the geometric view, the electromagnetic field is a six-dimensional geometric object in spacetime as opposed to two interdependent, but separate, 3-vector fields in space and time.
• Yeoh hyperelastic model — The Yeoh hyperelastic material model is a phenomenological model for the deformation of nearly incompressible, nonlinear elastic materials such as rubber.
• Μ(I) rheology — In granular mechanics, the \boldsymbol{\mu(I)} rheology is a rheological model for granular flows, describing the evolution of the macroscopic friction coefficient \mu as a function of the dimensionless quantity I called the Inertial number.